Everything about Time Scale Calculus totally explained
In
mathematics,
time scale calculus is a unification of the theory of
difference equations and standard
calculus. Discovered in
1988 by the German mathematician
Stefan Hilger, it has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if you differentiate a function which acts on the real numbers then the definition is equivalent to standard differentiation, but if you use a function acting on the integers then it's equivalent to the forward difference operator. A precise definition follows at the end of this article:
Dynamic equations
Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their
continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown
function is a so-called time scale, which may be an arbitrary closed subset of the reals. In this way, results not only related to the
set of
real numbers or set of
integers but those pertaining to more general time scales are obtained.
The three most popular examples of
calculus on time scales are
differential calculus,
difference calculus, and
quantum calculus. Dynamic equations on a time scale have a potential for applications, such as in
population dynamics. For example, it can model insect populations that are continuous while in season, die out in say winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population. Since then several authors have expounded on various aspects of this new theory.
Precise definition
A
time scale or
measure chain T is a
closed subset of the
real line R.
Define:
» σ(
t) = inf (backward shift operator)
Let
t be an element of
T:
t is:
» left dense if ρ(
t) =
t,
right dense if σ(
t) =
t,
» left scattered if ρ(
t) <
t,
right scattered if σ(
t) >
t,
» dense if left dense or right dense.
Define the
graininess μ of a measure chain
T by:
» μ(
t) = σ(
t) −
t.
Take a function:
» f :
T →
R,
(where R could be any
Banach space, but set it to be the real line for simplicity).
Definition:
generalised derivative or
fdelta(
t)
For every ε > 0 there exists a neighbourhood
U of
t such that:
» |
f(σ(
t)) −
f(
s) −
fdelta(
t)(σ(
t) −
s)| ≤ ε|σ(
t) −
s|
for all
s in
U.
Take
T =
R. Then σ(
t) =
t,μ(
t) = 0,
fdelta =
f′ is the derivative used in standard
calculus. If
T =
Z (the
integers), σ(
t) =
t + 1, μ(
t)=1,
fdelta = Δ
f is the
forward difference operator used in difference equations.
Further Information
Get more info on 'Time Scale Calculus'.
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